Triangles — Class 10 Mathematics NCERT Solutions (Free)

Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Triangles" — 9 important questions with detailed answers for CBSE board exam preparation.

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TL;DR: Free step-by-step NCERT solutions for Class 10 Mathematics chapter "Triangles" — 9 important questions with detailed answers for CBSE board exam prepa…

Written & reviewed by the Syllab.in Academic Team (CBSE/NCERT subject experts) · Updated Jul 23, 2026

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Key Questions Covered:

State and explain the Basic Proportionality Theorem (Thales' Theorem). What d…
Two triangles ABC and PQR are similar. If AB = 4 cm, BC = 6 cm, CA = 5 cm, an…
In triangle ABC, DE || BC where D is on AB and E is on AC. If AD = 3 cm, DB =…
Two similar triangles have corresponding sides in the ratio 3:5. If the area …
In triangle ABC, a line parallel to BC intersects AB at D and AC at E such th…
In a right triangle ABC with right angle at B, the altitude BD is drawn to th…
+ 3 more questions in the full chapter

Solutions Summary:

Question	Status
State and explain the Basic Proportionality Theorem (Thal…	✓ Solved
Two triangles ABC and PQR are similar. If AB = 4 cm, BC =…	✓ Solved
In triangle ABC, DE \|\| BC where D is on AB and E is on AC…	✓ Solved
Two similar triangles have corresponding sides in the rat…	✓ Solved
In triangle ABC, a line parallel to BC intersects AB at D…	✓ Solved
In a right triangle ABC with right angle at B, the altitu…	✓ Solved

Showing 6 of 9 questions

Q1: State and explain the Basic Proportionality Theorem (Thales' Theorem). What does it tell us about similar triangles?

Basic Proportionality Theorem (BPT): If a line is drawn parallel to one side of a triangle, then it divides the other two sides in the same ratio. Statement: In triangle ABC, if DE || BC (DE is parallel to BC), where D is on AB and E is on AC, then: AD/DB = AE/EC Explanation: - The parallel line DE creates two smaller triangles: ADE and ABC - These triangles are similar (AA similarity — angle A is common, and corresponding angles are equal due to parallel lines) - Similar triangles have propor...

Q2: Two triangles ABC and PQR are similar. If AB = 4 cm, BC = 6 cm, CA = 5 cm, and PQ = 6 cm, find QR and RP.

Given: Triangle ABC ~ Triangle PQR AB = 4 cm, BC = 6 cm, CA = 5 cm PQ = 6 cm Find: QR and RP Step 1: Since ABC ~ PQR, the corresponding sides are proportional. Corresponding sides are: AB ↔ PQ, BC ↔ QR, CA ↔ RP Step 2: Write the ratio of similarity. AB/PQ = BC/QR = CA/RP = k (ratio of similarity) Step 3: Find the ratio k. k = AB/PQ = 4/6 = 2/3 Step 4: Find QR. BC/QR = 2/3 6/QR = 2/3 QR = 6 × 3/2 = 9 cm Step 5: Find RP. CA/RP = 2/3 5/RP = 2/3 RP = 5 × 3/2 = 7.5 cm Answer: QR = 9 cm, RP = 7...

Q3: In triangle ABC, DE || BC where D is on AB and E is on AC. If AD = 3 cm, DB = 6 cm, and AE = 2 cm, find EC.

Given: Triangle ABC with DE || BC D is on AB, E is on AC AD = 3 cm, DB = 6 cm, AE = 2 cm Find: EC Step 1: Apply Basic Proportionality Theorem. Since DE || BC, we have: AD/DB = AE/EC Step 2: Substitute the known values. 3/6 = 2/EC Step 3: Solve for EC. 1/2 = 2/EC EC = 2 × 2 = 4 cm Answer: EC = 4 cm Verification: AD/DB = 3/6 = 1/2 and AE/EC = 2/4 = 1/2 ✓

Q4: Two similar triangles have corresponding sides in the ratio 3:5. If the area of the smaller triangle is 36 cm², find the area of the larger triangle.

Given: Two similar triangles with sides in ratio 3:5 Area of smaller triangle = 36 cm² Find: Area of larger triangle Step 1: Recall the relationship between areas of similar triangles. When two triangles are similar with sides in ratio k:1, their areas are in ratio k²:1. Step 2: Here the ratio of sides = 3:5 Ratio of areas = (3)²:(5)² = 9:25 Step 3: Let the area of the larger triangle be A. Area of smaller triangle : Area of larger triangle = 9:25 36:A = 9:25 Step 4: Solve for A. 36/A = 9/2...

Q5: In triangle ABC, a line parallel to BC intersects AB at D and AC at E such that AD:DB = 2:1. If BC = 15 cm, find DE.

Given: DE || BC in triangle ABC AD:DB = 2:1, BC = 15 cm Find: DE Step 1: Express AD and DB in terms of a variable. Let AD = 2x and DB = x Then AB = AD + DB = 2x + x = 3x Step 2: Find the ratio AD:AB. AD:AB = 2x:3x = 2:3 Step 3: Triangles ADE and ABC are similar (AA similarity). Since DE || BC: ∠ADE = ∠ABC (corresponding angles) ∠AED = ∠ACB (corresponding angles) ∠A is common Step 4: Write the ratio of corresponding sides. Since triangles ADE and ABC are similar: AD/AB = DE/BC = AE/AC Step ...

Q6: In a right triangle ABC with right angle at B, the altitude BD is drawn to the hypotenuse AC. Prove that AB² = AD × AC.

Given: Triangle ABC with ∠B = 90° BD is the altitude from B to hypotenuse AC Prove: AB² = AD × AC Proof: Step 1: Consider triangles ABD and ABC. In triangle ABD: ∠ADB = 90° (BD is perpendicular to AC) ∠BAD = ∠BAC (common angle) Therefore ∠ABD = ∠ABC - 90° Step 2: In triangle ABC: ∠ABC = 90° (given) ∠BAC = ∠BAC (common) ∠BCA = 90° - ∠BAC Step 3: Consider triangles ABD and CBA. In triangle ABD: ∠ADB = 90°, ∠BAD = ∠BAC In triangle CBA: ∠ABC = 90°, ∠BCA = ∠BCA Step 4: Show that triangle ABD ~...

Showing 6 of 9 questions. Visit the full page for complete solutions.